Counting Endomorphisms of Crown-like Orders

Families of fast elliptic curves from Q-curves

We construct new families of elliptic curves over Fp2 with efficiently computable endomorphisms, which can be used to accelerate elliptic curvebased cryptosystems in the sameway asGallant–Lambert–Vanstone (GLV) and Galbraith–Lin–Scott (GLS) endomorphisms. Our construction is based on reducingQ-curves—curves over quadratic number fields without complex multiplication, butwith isogenies to their ...

Families of Fast Elliptic Curves from ℚ-curves

We construct new families of elliptic curves over Fp2 with efficiently computable endomorphisms, which can be used to accelerate elliptic curvebased cryptosystems in the same way as Gallant–Lambert–Vanstone (GLV) and Galbraith–Lin–Scott (GLS) endomorphisms. Our construction is based on reducing Q-curves—curves over quadratic number fields without complex multiplication, but with isogenies to th...

Combinatorial problems from transformation semigroups

algebra and enumerative combinatorics We learn something when we can interpret combinatorial numbers as orders of algebraic structures, especially groups. Lagrange’s Theorem means that group orders tend to have a rich arithmetic structure. I first learned from Norman’s paper on chip-firing games the fact that the number of spanning trees of a graph is the order of an abelian group canonically a...

Compact Endomorphisms of Certain Subalgebras of the Disk Algebra

The Q-curve construction for endomorphism-accelerated elliptic curves

We give a detailed account of the use of Q-curve reductions to construct elliptic curves over Fp2 with efficiently computable endomorphisms, which can be used to accelerate elliptic curve-based cryptosystems in the same way as Gallant–Lambert–Vanstone (GLV) and Galbraith–Lin–Scott (GLS) endomorphisms. Like GLS (which is a degenerate case of our construction), we offer the advantage over GLV of ...